Totally-disconnected space

A topological space in which any subset containing more than one point is disconnected (cf. connected space). An equivalent condition is that the connected component of any point in the space is that point itself. The topological product and the topological sum of totally-disconnected spaces, as well as any subspace of a totally-disconnected space, are totally disconnected. Any totally-disconnected compactum is a zero-dimensional space (in every sense). Such compacta are important, in particular, because they are Stone spaces of Boolean algebras. A totally-disconnected space (a Knaster–Kuratowski fan) in the plane that can be made into a connected space as a result of the addition of a single point, has been constructed. Such a space is not zero-dimensional. In a Hilbert space, the subspace formed by the points all coordinates of which are rational is totally disconnected and one-dimensional. If each point in a space is the intersection of all closed-and-open sets including it, the space is totally disconnected (in particular, all zero-dimensional spaces are totally disconnected). However, there exists a totally-disconnected metric space with a countable base in which not all points are intersections of such closed-and-open sets.

References

[1]	W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)
[2]	R. Engelking, "Outline of general topology" , North-Holland (1968) (Translated from Polish)
[3]	J.L. Kelley, "General topology" , Springer (1975)
[4]	N. Bourbaki, "Elements of mathematics. General topology" , Springer (1989) (Translated from French)

Comments

There is a totally-disconnected plane set $E$, no proper superset of which is totally disconnected, [a3]. The complement of such a set is called a primitive dispersion set for the plane. For all $n$, there are $n$-dimensional totally-disconnected separable metric groups, [a4].

There is some confusion in the terminology concerning disconnected spaces. There are a few kinds of disconnectedness; the two most common notions are: i) the one in the article: connected subsets consist of at most one point; and ii) for every two points $x$ and $y$ there is a closed-and-open set $C$ such that $x\in C$ and $y\notin C$.

Both are called total disconnectedness at times. References [a1] and [a2] call spaces satisfying ii) totally disconnected, and in [a2] spaces satisfying i) are called hereditarily disconnected (because they have no non-trivial connected subspaces). (Note that ii) implies i).)

The Knaster–Kuratowski fan is a subset of the plane defined as follows: Consider the usual Cantor middle-third set $C$ situated in the interval $[0,1]\times\{0\}$ in the plane. Connect every point $x$ of $C$ with the point $(1/2,1/2)$ by the line segment $L_x$. For each $x\in C$ take a subset $F_x$ of $L_x$ as follows: if $x$ is an end-point of an interval in the complement of $C$ take all points of $L_x$ with rational second coordinate, otherwise take the points with irrational second coordinate. The union $\bigcup_{x\in C}F_x$ is the Knaster–Kuratowksi fan. When one removes the point $(1/2,1/2)$ from $F$, one obtains a space that satisfies i) above but not ii). See also Kuratowski–Knaster fan.

References

[a1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[a2]	R. Engelking, "General topology" , Heldermann (1989)
[a3]	M.E. Estill, "A primitive dispersion set of the plane" Duke Math. J. , 9 : 19 (1952) (323–328)
[a4]	J. van Mill, "$n$-dimensional totally disconnected topological groups" Math. Japon. , 32 (1987) pp. 267–273